IPMAT Rohtak 2019 (QA) PYQs

For $0\lt\theta\lt\frac{\pi}{4}$, let $a=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \cos \theta\right), b=\left((\cos \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right), c=\left((\sin \theta)^{\cos \theta}\right)\left(\log _{2} \cos \theta\right)$ and $d=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$. Then, the median value in the sequence $a, b, c, d$ is

If a pole 6 m high casts a shadow $2\sqrt{3}$ m long on the ground, then the sun's elevation is

Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, then the distance between their tops is